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Graph Theory
Open
Seymour's Second Neighborhood Conjecture
In every oriented graph, is there a vertex whose out-neighborhood's second out-neighborhood is at least as large as its first? A clean statement about local expansion in orientations.
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- Validated increment
Settled for tournaments; open for general orientations
The conjecture is known for tournaments and several sparse or locally restricted oriented classes. The general oriented-graph case remains the target.
Previous best
Open even for tournaments (historical)
New certified
Proved for tournaments; open in general
Public literature baseline
Also open